The proof of Lemma REF  is given in Appendix . The basic outline of the proof is as follows: the fraction \(\frac{\vert \mathcal {S}_a\left(r\right) \cap \mathcal {B}_b\left(r\right)\vert }{\vert \mathcal {S}_a\left(r\right) \vert }\)  is at least the sum of probabilities from a hypergeometric distribution \(P_{k,m,r}\)  from \(\frac{m}{2}\)  to \(m\)  where \(m = \frac{k}{\log {k}}\)  is the distance between the Hamming Ball and the Sphere. We show in Lemma REF  that the hypergeometric distribution \(P_{k,m,r}\)  is symmetric about \(\frac{m}{2}\)  for a range up to \(200\sqrt{m}\) . The expected value \(E\)  of \(P_{k,m,r}\)  for our choice of \(m\)  and \(r\)  lies between \(\frac{m}{2} - 100\sqrt{m}\)  and \(\frac{m}{2} - \sqrt{m}\) . We have a concentration bound for hypergeometric distribution \(P_{k,m,r}\)  by Hoeffding [1]} stated in Lemma REF  that the sum of the probabilities around the expected value of width \(\sqrt{r}\)  is at least 0.7. Using the property of hypergeometric distributions that it is monotone increasing up to the expected value \(E\)  and monotone decreasing beyond it shown in Lemma REF , we derive a concentration bound of width \(\sqrt{m}\)  around \(E\)  that the probabilities in this range sum to at least \(0.7 \times \frac{\sqrt{m}}{\sqrt{r}}\) , which for our choice of \(m\)  and \(r\)  is at least \(\frac{1}{\sqrt{\log {k}}}\) . This gives us that the \(\frac{\vert \mathcal {S}_a\left(r\right) \cap \mathcal {B}_b\left(r\right)\vert }{\vert \mathcal {S}_a\left(r\right) \vert } \ge \frac{c^{\prime }}{\sqrt{\log {k}}}\)  for a constant \(c^{\prime }\) .
